Question: Suppose that points $P_1$, $P_2$, and $P_3$ are chosen uniformly at random on the sides of a square $T$. Compute the probability that $$\frac{[\triangle P_1 P_2 P_3]}{[T]}>\frac{1}{4}$$ where $[X]$ denotes the area of polygon $X$.
Without loss of generality, I assumed the side length of the square to be $1$. Because the question mentions $\frac{1}{4}$ of the square, I considered splitting the square into quadrants.
- It is obvious that all three points cannot lie in the same quadrant of the square.
- Similarly, there cannot be two points in the same quadrant because the area must be less than $\frac{1}{2} \cdot \frac{1}{2} \cdot 1=\frac{1}{4}$. [EDIT: This is wrong]
This means that all three vertices must lie in different quadrants in the square. From here, I considered cases:
Case 1: Two of the points are on the same side, which occurs with probability $\frac{9}{16}$.
Case 2: All three points are on different sides, which occurs with probability $\frac{3}{8}$.
From here, my efforts have consisted of just labeling lengths and finding the area in terms of said lengths. However, this has led me with some inequalities that I don't know how to find the probabilities of being true, namely:
- $x-xz+yz<\frac{1}{2}$ for $0 \leq x,y,z \leq 1$
- $(x-y)z<\frac{1}{2}$ fo $0 \leq y<x \leq 1$ and $0 \leq x \leq 1$
If anyone knows how to either find the probability of these inequalities being satisfied (which I think requires multivariable calculus) or a way that circumvents these inequalities, please let me know.
Here's my full attempt at the question, though I'm unsure about the accuracy of how I find the probabilities of the three-variable inequalities being satisfied. [NOTE: this method has a few errors in it, a correct version is in the answers below.]
Without loss of generality, consider the square with vertices $(0,0), (0,1), (1,0), (1,1)$.
We proceed using casework:
- All three vertices are on different sides of the square. This occurs with probability $\frac{9}{16}$.
Let the vertices be $(x_1,0)$, $(0,y_1)$, and $(x_2,1)$. Using the determinant form for the area of a triangle, the area is given by $$A=\frac{1}{2} \begin{vmatrix} x_1 & 0 & 1 \\ 0 & y_1 & 1 \\ x_2 & 1 & 1 \end{vmatrix}=\frac{1}{2}x_1-\frac{1}{2}y_1 (x_1-x_2)$$
Assume that $x_1>x_2$. Then, $$\frac{1}{2}x_1-\frac{1}{2}y_1 (x_1-x_2)>\frac{1}{4} \implies x_1-y_1 (x_1-x_2)>\frac{1}{2}$$
For convenience, replace $x_1$ with $y$, $x_2$ with $x$, and $y_1$ with $z$. We now have the system of inequalities $\begin{cases} y-yz+xz>\frac{1}{2} \\ y>x \\ 0 \leq x,y,z \leq 1 \end{cases}$.
We now consider graphing the inequality $y-yz+xz>\frac{1}{2}$ with $z$ as a constant. The $x$-intercept is $\frac{1}{2z}$ and the $y$-intercept is $\frac{1}{2(1-z)}$.
- If $0<z \leq \frac{1}{2}$, then the area of the region is given by $\frac{4z-1}{8z-8}$. Therefore, the desired probability for this case is $\int^{\frac{1}{2}}_{0} \frac{4z-1}{8z-8} \, dz=\frac{1}{8}(2-\ln 2)$.
- If $\frac{1}{2} \leq z<1$, then the area of the region is given by $\frac{1}{8z}$. Therefore, the desired probability for this case is $\int^{\frac{1}{2}}_{0} \frac{1}{8z}=\frac{1}{8} \ln 2$.
The overall probability for this case is $$\frac{9}{16} \cdot \frac{1}{2} \cdot \frac{1}{4}=\frac{9}{128}$$
- Two vertices lie on the same side of the square and the third vertex lies on the opposite side. This occurs with probability $\frac{1}{8}$.
Let the vertices be $(y,1)$, $(x,1)$, and $(z,0)$ where $y>x$. Then, the area is given by $\frac{1}{2}(y-x)$, meaning we need $y-x>\frac{1}{2}$. It is easy to find the probability for this case is $$\frac{1}{8} \cdot \frac{1}{8}=\frac{1}{64}$$
- Two vertices lie on the same side of the square and the third vertex lies on an adjacent side. This occurs with probability $\frac{1}{4}$.
Let the vertices of the triangle be $(x,1)$, $(y,1)$, and $(1,1-z)$ where $y>x$. Then, the area is given by $\frac{1}{2}(y-x)(z)$, meaning we need $(y-x)z>\frac{1}{2}$.
It is easy to see that this is only possible for $\frac{1}{2} \leq z <1$, in which case the probability is $\frac{2z-1}{8z^2}$. The probability for this case is thus $$\frac{1}{4} \cdot \int^{1}_{\frac{1}{2}} \frac{2z-1}{8z^2} \, dz=\frac{1}{64} \ln 4-\frac{1}{64}$$
Therefore, the final answer is $$\frac{1}{64} \ln 4-\frac{1}{64}+\frac{1}{64}+\frac{9}{128}=\boxed{\frac{9+4 \ln 2}{128}}$$